Found problems: 5802
For any positive integer $b\ge2$, we write the base-$b$ numbers as follows:
\[(d_kd_{k-1}\dots d_0)_b=d_kb^k+d_{k-1}b^{k-1}+\dots+d_1b^1+d_0b^0,\]where each digit $d_i$ is a member of the set $S=\{0,1,2,\dots,b-1\}$ and either $d_k\not=0$ or $k=0$. There is a unique way to write any nonnegative integer in the above form. If we select the digits from a di fferent set $S$ instead, we may obtain new representations of all positive integers or, in some cases, all integers. For example, if $b=3$ and the digits are selected from $S=\{-1,0,1\}$, we obtain a way to uniquely represent all integers, known as a $\emph{balanced ternary}$ representation. As further examples, the balanced ternary representation of numbers $5$, $-3$, and $25$ are:
\[5=(1\ {-1}\ {-1})_3,\qquad{-3}=({-1}\ 0)_3,\qquad25=(1\ 0\ {-1}\ 1)_3.\]However, not all digit sets can represent all integers. If $b=3$ and $S=\{-2,0,2\}$, then no odd number can be represented. Also, if $b=3$ and $S=\{0,1,2\}$ as in the usual base-$3$ representation, then no negative number can be represented.
Given a set $S$ of four integers, one of which is $0$, call $S$ a $\emph{4-basis}$ if every integer $n$ has at least one representation in the form
\[n=(d_kd_{k-1}\dots d_0)_4=d_k4^k+d_{k-1}4^{k-1}+\dots+d_14^1+d_04^0,\]where $d_k,d_{k-1},\dots,d_0$ are all elements of $S$ and either $d_k\not=0$ or $k=0$.
[list=a]
[*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is not a $4$-basis.
[*]Show that there are infinitely many integers $a$ such that $\{-1,0,1,4a+2\}$ is a $4$-basis.[/list]
Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$
Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$
Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$.
[hide=example]For $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )[/hide]
For a positive integer $n$, $n$ vertices which have $10000$ written on them exist on a plane. For $3$ vertices that are collinear and are written positive numbers on them, denote procedure $P$ as subtracting $1$ from the outer vertices and adding $2023$ to the inner vertical. Show that procedure $P$ cannot be repeated infinitely.
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win?
[i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]
Prove that for every positive integer $t$ there is a unique permutation $a_0, a_1, \ldots , a_{t-1}$ of $0, 1, \ldots , t-1$ such that, for every $0 \leq i \leq t-1$, the binomial coefficient $\binom{t+i}{2a_i}$ is odd and $2a_i \neq t+i$.
Let $A\in M_4(C)$ be a non-zero matrix.
$a)$ If $\text{rank}(A)=r<4$, prove the existence of two invertible matrices $U,V\in M_4(C)$, such that:
\[UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix}\]
where $I_r$ is the $r$-unit matrix.
$b)$ Show that if $A$ and $A^2$ have the same rank $k$, then the matrix $A^n$ has rank $k$, for any $n\ge 3$.
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link [i]good[/i], if it touches $\omega$, and [i]bad[/i] otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks a integer $c \in \{2,3,\dots,p-1\}$.
Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper.
Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing.
The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
[i]Proposed by Yang Liu[/i]
A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$.
Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?)
[i](Folklore)[/i]
A rectangular array has 9 rows and 2004 columns. In the 9 * 2004 cells of the table we place the numbers from 1 to 2004, each 9 times. And we do this in such a way that two numbers, which stand in exactly the same column in and differ around at most by 3. Find the smallest possible sum of all numbers in the first row.
Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$(Here $a_{0}=a_{n},a_{n+1}=a_{1}$). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$.
Let be $n$ positive integer than calculate:
$1\cdot 1!+2\cdot2!+...+n\cdot n!$
The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities
$a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$
$a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$
Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$
Let $m_1< m_2 < \ldots m_{k-1}< m_k$ be $k$ distinct positive integers such that their reciprocals are in arithmetic progression.
1.Show that $k< m_1 + 2$.
2. Give an example of such a sequence of length $k$ for any positive integer $k$.
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
Let $ f(n)$ be defined for $ n \in \mathbb{N}$ by $ f(1)\equal{}2$ and $ f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1$ for $ n \ge 1$. Prove that for all $ n >1:$
$ 1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}$
Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.
Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards $1, 3, 5,7,..., 2n-1$
b) Bruno has cards $2, 4,6, 8,...,2n$
During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after $n$ turns. Determine the highest number of points Ana can earn, and how she manages to do this.
Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.
There is a row of $100$ cells each containing a token. For $1$ dollar it is allowed to interchange two neighbouring tokens. Also it is allowed to interchange with no charge any two tokens such that there are exactly $3$ tokens between them. What is the minimum price for arranging all the tokens in the reverse order?
(Egor Bakaev)
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\]
(A disk is assumed to contain its boundary.)